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Francisco M. Fernández

Publications and source records attributed to Francisco M. Fernández.

At least 109 records · Page 6Linked to original sources
arXiv2017

Exceptional points in two simple textbook examples

We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well known damped harmonic oscillator. They enable one to connect the occurrence of linearly dependent exponential solutions with a defective matrix that cannot be diagonalized but can be transformed into a Jordan canonical form.

physics.class-ph↗
arXiv2017

Born-Oppenheimer approximation for a harmonic molecule

We apply the Born--Oppenheimer approximation to a harmonic diatomic molecule with one electron. We compare the exact and approximate results not only for the internal degrees of freedom but also for the motion of the center of mass. We address the problem of the permutation symmetry of identical nuclei and discuss other applications of the model and its limitations.

math-ph↗
arXiv2017

Application of the Kronecker product to simple spin systems

We show that the well known Kronecker product is a suitable tool for the construction of matrix representations of widely used spin Hamiltonians. In this way we avoid the explicit use of basis sets for the construction of the matrix elements. As illustrative examples we discuss two isotropic models and an anisotropic one.

quant-ph↗
arXiv2017

A pedagogical description of diabatic and adiabatic molecular processes

We provide a pedagogical approach to the problem of avoided crossings between electronic molecular curves and to diabatic and adiabatic transitions when the nuclei of a diatomic molecule move according to classical mechanics. For simplicity we restrict the analysis to only two electronic states.

quant-ph↗
arXiv2017

A quantum-mechanical anharmonic oscillator with a most interesting spectrum

We revisit the problem posed by an anharmonic oscillator with a potential given by a polynomial function of the coordinate of degree six that depends on a parameter $λ$. The ground state can be obtained exactly and its energy $E_{0}=1$ is independent of $λ$. This solution is valid only for $λ>0$ because the eigenfunction is not square integrable otherwise. Here we show that the perturbation series for the expectation values are Padé and Borel-Padé summable for $λ>0$. When $% λ<0$ the spectrum exhibits an infinite number of avoided crossings at each of which the eigenfunctions undergo dramatic changes in their spatial distribution that we analyze by means of the expectation values $\langle x^{2}\rangle $.

quant-ph↗
arXiv2016

Perturbation theory for short-range weakly-attractive potentials in one dimension

We have obtained the perturbative expressions up to sixth order for the energy of the bound state in a one dimensional, arbitrarily weak, short range finite well, applying a method originally developed by Gat and Rosenstein Ref. [3]. The expressions up to fifth order reproduce the results already known in the literature, while the sixth order had not been calculated before. As an illustration of our formulas we have applied them to two exactly solvable problems and to a nontrivial problem.

quant-ph↗
arXiv2016

On two perturbation series for short-range attractive potentials

We compare two alternative expansions for finite attractive wells. One of them is known from long ago and is given in terms of powers of the strength parameter. The other one is based on the solution of the equations of the Rayleigh-Schrödinger perturbation theory in a basis set of functions of period $L$. The analysis of exactly solvable models shows that although the exact solution of the problem with periodic boundary conditions yields the correct result when $L\rightarrow \infty$ the coefficients of the series for this same problem blow up and fail to produce the correct asymptotic expansion.

quant-ph↗
arXiv2016

On the symmetry of four particles in a one-dimensional box with harmonic interaction

We show that a system of four particles in a one-dimensional box with a two-particle harmonic interaction can by described by means of the symmetry point group $O_h$. Group theory proves useful for the discussion of both the small-box and large-box regimes. We apply perturbation theory and obtain the corrections of first order for the lowest states. We carry out a simple Rayleigh-Ritz variational calculation with basis sets adapted to the symmetries of the system. We also obtain alternative variational results for the first three lowest energy levels that are more suitable for larger box sizes.

quant-ph↗
arXiv2016

Bound states in weakly deformed waveguides: numerical vs analytical results

We have studied the emergence of bound states in weakly deformed and/or heterogeneous waveguides, comparing the analytical predictions obtained using a recently developed perturbative method, with precise numerical results, for different configurations (a homogeneous asymmetric waveguide, a heterogenous asymmetric waveguide and a homogeneous broken-strip). In all the examples considered in this paper we have found excellent agreement between analytical and numerical results, thus providing a numerical verification of the analytical approach.

physics.comp-ph↗
arXiv2016

On two different kinds of resonances in one-dimensional quantum-mechanical models

We apply the Riccati-Padé method and the Rayleigh-Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh-Ritz method reveals each set at a particular interval of rotation angles the Riccati Padé method yields both of them as roots of the same Hankel determinants.

quant-ph↗
arXiv2016

Generalization of parity-time and partial parity-time symmetry

We show that parity-time and partial parity-time symmetries are particular cases of antiunitary symmetry. This point is illustrated by means of a recently discussed system of non-Hermitian coupled harmonic oscillators that also exhibits other types of antiunitary symmetries. We also show that a combination of group and perturbation theory is a useful tool for predicting broken antiunitary symmetry.

quant-ph↗
arXiv2016

Simple model for Crystal Field Theory

We investigate a simple model for the prediction of the splitting of the $3d$ orbitals of a metal ion in the environment of ligands. The electrons are considered to be independent and their interaction with the ligands is represented by the Dirac delta function in three dimensions. We discuss several cases where the model is successful and a few ones where it is not.

cond-mat.mtrl-sci↗
arXiv2016

Improved power-series method for confined one-dimensional quantum-mechanical problems

We propose two improvements to the well-known power series method for confined one-dimensional quantum-mechanical problems. They consist of the addition of a variational step were the energy plays the role of a variational parameter. We compare the rate of convergence of the three methods on an exactly-solvable model. We also outline possible generalizations of the approaches to more complex problems.

quant-ph↗
arXiv2015

Algebraic treatment of a simple model for the electromagnetic self-force

The problem of the electromagnetic self-force can be studied in terms of a quadratic PT-symmetric Hamiltonian. Here, we apply a straightforward algebraic method to determine the regions of model-parameter space where the quantum-mechanical operator exhibits real spectrum. An alternative point of view consists of finding the values of the model parameters so that a symmetric operator supports bound states.

quant-ph↗